\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 165, pp. 1--6.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/165\hfil Multiple solutions] {Multiple solutions for quasilinear elliptic problems with nonlinear boundary conditions} \author[N. T. Chung\hfil EJDE-2008/165\hfilneg] {Nguyen Thanh Chung} \address{Department of Mathematics and Informatics\\ Quang Binh University\\ 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam} \email{ntchung82@yahoo.com} \thanks{Submitted October 20, 2008. Published December 23, 2008.} \subjclass[2000]{35J65, 35J20} \keywords{Multiple solutions; quasilinear elliptic problems; \hfill\break\indent nonlinear boundary conditions} \begin{abstract} Using a recent result by Bonanno \cite{2}, we obtain a multiplicity result for the quasilinear elliptic problem \begin{gather*} - \Delta_p u + |u|^{p-2}u = \lambda f(u) \quad \text{in } \Omega, \\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu} = \mu g(u) \quad \text{on } \partial\Omega, \end{gather*} where $\Omega$ is a bounded domain in $\mathbb R^N$, $N \geq 3$ with smooth boundary $\partial\Omega$, $\frac{\partial}{\partial\nu}$ is the outer unit normal derivative, the functions $f, g$ are $(p-1)$-sublinear at infinity ($1 0$ such that for all $t \in \mathbb{R}^N$, $$|f(t)| \leq M_1(1+|t|^{p-1}), \quad |g(t)| \leq M_2|t|^{p-1}\,;$$ \item[(H2)] $f$ is superlinear at zero; i.e., $$\lim_{t\to 0}\frac{f(t)}{|t|^{p-1}} = 0;$$ \item[(H3)] if we set $F(t) = \int_0^t f(t)dt$ and $G(t) = \int_0^t g(t)dt$, then there exists $t_0 \in \mathbb{R}$ such that $$F(t_0) = \int_0^{t_0}f(t)dt > 0 \quad\text{or}\quad G(t_0) = \int_0^{t_0}g(t)dt > 0.$$ \end{enumerate} Let $W^{1,p}(\Omega)$ be the usual Sobolev space with respect to the norm $$\|u\|_{1,p}^p = \int_\Omega (|\nabla u|^p + |u|^p) dx$$ and $W^{1,p}_0(\Omega)$ the closure of $C^\infty_0(\Omega)$ in $W^{1,p}(\Omega)$. For any $1 < p < N$ and $1 \leq q \leq p^\star = \frac{Np}{N-p}$, we denote by $S_{q,\Omega}$ the best constant in the embedding $W^{1,p}(\Omega) \hookrightarrow L^q(\Omega)$ and for all $1 \leq q \leq p_\star = \frac{(N-1)p}{N-p}$, we also denote by $S_{q,\partial\Omega}$ the best constant in the embedding $W^{1,p}(\Omega) \hookrightarrow L^q(\partial\Omega)$, i.e. $$S_{q,\partial\Omega} = \inf_{u \in W^{1,p}(\Omega)\backslash W^{1,p}_0(\Omega)} \frac{\int_\Omega (|\nabla u|^p + |u|^p) dx}{\big(\int_{\partial\Omega} |u|^qd\sigma\big)^{p/q}}.$$ Moreover, if $1 \leq q < p^\star$, then the embedding $W^{1,p}(\Omega) \hookrightarrow L^q(\Omega)$ is compact and if $1 \leq q < p_\star$, then the embedding $W^{1,p}(\Omega) \hookrightarrow L^q(\partial\Omega)$ is compact. As a consequence, we have the existence of extremals, i.e. functions where the infimum is attained (see \cite{3, 6}). \begin{definition}\label{def1.1} \rm We say that $u \in W^{1,p}(\Omega)$ is a weak solution of problem (\ref{e1.1}) if and only if $$\int_\Omega (|\nabla u|^{p-2}\nabla u\nabla\varphi + |u|^{p-2}u\varphi)dx - \lambda \int_\Omega f(u)\varphi dx - \mu \int_{\partial\Omega} g(u) \varphi d\sigma = 0$$ for all $\varphi \in W^{1,p}(\Omega)$. \end{definition} \begin{theorem}\label{thm1.2} Assuming hypotheses {\rm (H1)--(H3)} are fulfilled then there exist an open interval $\Lambda_\mu$ and a constant $\delta_\mu >0$ such that for all $\lambda \in \Lambda_\mu$, problem (\ref{e1.1}) has at least two weak solutions in $W^{1,p}(\Omega)$ whose $\|\cdot\|_{1,p}$-norms are less than $\delta_\mu$. \end{theorem} We emphasize that the condition (H3) cannot be omitted. Indeed, if for instance $f\equiv 0$ and $g\equiv 0$, then (H1) and (H2) clearly hold, but problem (\ref{e1.1}) has only the trivial solution. Theorem \ref{thm1.2} will be proved by using a recent result on the existence of at least three critical points by Bonanno \cite{2} which is actually a refinement of a general principle of Ricceri (see \cite{9, 10}). For the reader's convenience, we describe it as follows. \begin{lemma}[see {\cite[Theorem 2.1]{2}}] \label{lem1.3} Let $(X, \|\cdot\|)$ be a separable and reflexive real Banach space, $\mathcal A, \mathcal F : X \to \mathbb{R}$ be two continuously G\^{a}teaux differentiable functionals. Assume that there exists $x_0 \in X$ such that $\mathcal A(x_0) = \mathcal F(x_0) = 0$, $\mathcal A(x) \geq 0$ for all $x \in X$ and there exist $x_1 \in X$, $\rho > 0$ such that \begin{itemize} \item[(i)] $\rho < \mathcal A(x_1)$, \item[(ii)] $\sup_{\{\mathcal A(x)< \rho\}} \mathcal F(x) < \rho \frac{\mathcal F(x_1)}{\mathcal A(x_1)}$. \end{itemize} Further, put $$\overline{a} = \frac{\xi \rho}{\rho \frac{\mathcal F(x_1)}{\mathcal A(x_1)} - \sup_{\{\mathcal A(x) < \rho\}}\mathcal F(x)}, \quad \text{with } \xi > 1,$$ and assume that the functional $\mathcal A - \lambda \mathcal F$ is sequentially weakly lower semicontinuous, satisfies the Palais-Smale condition and \begin{itemize} \item[$(iii)$] $\lim_{\|x\| \to \infty}[\mathcal A(x) - \lambda \mathcal F(x)] = +\infty$ for every $\lambda \in [0, \overline{a}]$. \end{itemize} Then, there exist an open interval $\Lambda \subset [0,\overline{a}]$ and a positive real number $\delta$ such that each $\lambda \in \Lambda$, the equation $$D\mathcal A(u) - \lambda D\mathcal F(u) = 0$$ has at least three solutions in $X$ whose $\|\cdot\|$-norms are less than $\delta$. \end{lemma} \section{Multiple solutions} Throughout this section, we suppose that all assumptions of Theorem \ref{thm1.2} are satisfied. For $\lambda$ and $\mu \in \mathbb{R}$, we define the functional $\Phi_{\mu,\lambda} : W^{1,p}(\Omega) \to \mathbb{R}$ by $$\Phi_{\mu,\lambda}(u) = \mathcal I_\mu(u) - \lambda \mathcal J (u) \text{ for all } u \in W^{1,p}_0(\Omega),$$ where $$\label{e2.1} \mathcal I_\mu(u) = \int_\Omega (|\nabla u|^p + |u|^p)dx - \mu \int_{\partial\Omega} G(u) d\sigma, \quad \mathcal J (u) = \int_\Omega F(u) dx$$ with $F(t) = \int_0^t f(t)dt$ and $G(t) = \int_0^t g(t)dt$. A simple computation implies that the functional $\Phi_{\mu,\lambda}$ is of $C^1$-class and hence weak solutions of (\ref{e1.1}) correspond to the critical points of $\Phi_{\mu,\lambda}$. To prove Theorem \ref{thm1.2}, we shall apply Lemma \ref{lem1.3} by choosing $X = W^{1,p}(\Omega)$ as well as $\mathcal A = \mathcal I_\mu$ and $\mathcal F = \mathcal J$ as in (\ref{e2.1}). Now, we shall check all assumptions of Lemma \ref{lem1.3}. For each $\mu \in [0, \frac{pS_{p,\partial\Omega}}{M_2})$ we have $\mathcal I_\mu (u) \geq 0$ for all $u \in W^{1, p}(\Omega)$ and $\mathcal I_\mu(0) = \mathcal J(0) = 0$ since the assumption (H1) holds. Moreover, by the compact embeddings $W^{1,p}(\Omega) \hookrightarrow L^p(\Omega)$ and $W^{1,p}(\Omega) \hookrightarrow L^p(\partial\Omega)$, a simple computation helps us to conclude the following lemma. \begin{lemma}\label{lem2.1} For every $\mu \in [0, \frac{pS_{p,\partial\Omega}}{M_2})$ and all $\lambda \in \mathbb{R}$, the functional $\Phi_{\mu,\lambda}$ is sequentially weakly lower semicontinuous on $W^{1,p}(\Omega)$. \end{lemma} \begin{lemma}\label{lem2.2} There exist two positive constants $\overline{\mu}$ and $\overline{\lambda}$ such that for all $\mu \in [0, \overline{\mu})$ and all $\lambda \in [0, \overline{\lambda})$, the functional $\Phi_{\lambda, \mu}$ is coercive and satisfies the Palais-Smale condition in $W^{1,p}(\Omega)$. \end{lemma} \begin{proof} By (H1), we have \begin{align}\label{e2.2} \Phi_{\mu, \lambda} (u) & = \int_\Omega (|\nabla u|^p + |u|^p)dx - \lambda \int_\Omega F(u) dx - \mu\int_{\partial\Omega} G(u) d\sigma\notag\\ & \geq \|u\|^p_{1,p} - \lambda M_1\int_\Omega (|u|+\frac{|u|^p}{p}) dx - \mu \frac{M_2}{p}\int_{\partial\Omega}|u|^pd\sigma\notag\\ & \geq \|u\|^p_{1,p}\Big(1 - \lambda\frac{M_1}{pS_{p, \Omega}} - \mu \frac{M_2}{pS_{p, \partial\Omega}}\Big) - \lambda \frac{M_1}{S_{1,\Omega}}\|u\|_{1,p}. \end{align} Since relation (\ref{e2.2}) holds, by choosing $$\overline{\mu} = \overline{\lambda} = \min\big\{\frac{pS_{p, \Omega}}{2M_1}, \frac{pS_{p, \partial\Omega}}{2M_2}\big\},$$ where $M_1$, $M_2$ are given in (H1), we conclude that for all $\lambda \in [0, \overline{\lambda})$ and all $\mu \in [0, \overline{\mu})$, the functional $\Phi_{\mu,\lambda}$ is coercive. Now, let $\{u_m\}$ be a Palais-Smale sequence for the functional $\Phi_{\mu, \lambda}$ in $W^{1,p}(\Omega)$; i.e., $$\label{e2.3} |\Phi_{\mu,\lambda}(u_m)| \leq \overline{c},\quad D\Phi_{\mu,\lambda}(u_m) \to 0 \text{ in } W^{-1,p}(\Omega),$$ where $W^{-1,p}(\Omega)$ is the dual space of $W^{1,p}(\Omega)$. Since $\Phi_{\mu,\lambda}$ is coercive, the sequence $\{u_m\}$ is bounded in $W^{1,p}(\Omega)$. Therefore, there exists a subsequence of $\{u_m\}$, denoted by $\{u_m\}$ such that $\{u_m\}$ converges weakly to some $u \in W^{1,p}(\Omega)$ and hence converges strongly to $u$ in $L^p(\Omega)$ and in $L^p(\partial\Omega)$. We shall prove that $\{u_m\}$ converges strongly to $u$ in $W^{1,p}(\Omega)$. Indeed, we have \begin{align*} \|u_m -u\|^p_{1,p} & \leq \int_\Omega (|\nabla u_m|^{p-2}\nabla u_m- |\nabla u|^{p-2}\nabla u)(\nabla u_m - \nabla u) dx\\ & \quad + \int_\Omega (|u_m|^{p-2}u_m - |u|^{p-2}u)(u_m-u)dx\\ & = [D\Phi_{\mu,\lambda}(u_m)-D\Phi_{\mu,\lambda}(u)](u_m-u) + \lambda \int_\Omega[f(u_m)-f(u)](u_m-u)dx \\ & \quad +\mu \int_{\partial\Omega}[g(u_m)-g(u)](u_m-u)dx. \end{align*} On the other hand, the compact embeddings and (H1) imply \begin{align*} &|\int_\Omega [f(u_m)-f(u)](u_m-u)dx| \\ & \leq \int_\Omega |f(u_m)-f(u)||u_m-u|dx \\ & \leq M_1 \int_\Omega (2 + |u_m|^{p-1}+|u|^{p-1})|u_m-u|dx \\ & \leq M_1(2 \mathop{\rm meas}(\Omega)^{\frac{p-1}{p}} +\|u_m\|^{p-1}_{L^p(\Omega)}+ \|u\|^{p-1}_{L^p(\Omega)})\|u_m-u\|^{p}_{L^p(\Omega)} \end{align*} which approaches $0$ as $m \to \infty$. Similarly, we obtain \begin{align*} |\int_{\partial\Omega} [g(u_m)-g(u)](u_m-u)dx| & \leq \int_{\partial\Omega} |g(u_m)-g(u)||u_m-u|dx \\ & \leq M_2 \int_{\partial\Omega} (|u_m|^{p-1}+|u|^{p-1})|u_m-u|dx \\ & \leq M_2(\|u_m\|^{p-1}_{L^p(\partial\Omega)}+ \|u\|^{p-1}_{L^p(\partial\Omega)})\|u_m-u\|^{p}_{L^p(\partial\Omega)} \end{align*} which approaches zero as $m \to \infty$. Hence, by (\ref{e2.3}) we have $\|u_m - u\|_{1,p} \to 0$ as $m \to \infty$; i.e., the functional $\Phi_{\mu,\lambda}$ satisfies the Palais-Smale condition. \end{proof} \begin{lemma}\label{lem2.3} For every $\mu \in [0, \overline{\mu})$ with $\overline{\mu}$ as in Lemma \ref{lem2.2}, we have $$\lim_{\rho \to 0^+}\frac{\sup\{\mathcal J (u) : \mathcal I_\mu (u) < \rho\}}{\rho} = 0.$$ \end{lemma} \begin{proof} Let $\lambda \in [0, \overline{\lambda})$ and $\mu \in [0, \overline{\mu})$ be fixed. By (H2), for any $\epsilon> 0$, there exists $\delta = \delta(\epsilon) > 0$ such that $$|f(s)| < \epsilon pS_{p, \Omega}\Big(1 - \mu\frac{M_2}{pS_{p, \partial\Omega}}\Big)|s|^{p-1} \text{ for all } |s| < \delta.$$ We first fix $q \in (p, p^\star)$. Combining the above inequalities with (H1) we deduce that $$\label{e2.4} |F(s)| \leq \epsilon S_{p, \Omega}\Big(1 - \mu\frac{M_2}{pS_{p, \partial\Omega}}\Big)|s|^{p}+ C_{\delta} |s|^q,$$ for all $s \in \mathbb{R}$, where $C_{\delta}$ is a constant depending on $\delta$. Now, for every $\rho > 0$, we define the sets $$\mathcal{B}^1_\rho = \{u \in W^{1,p}(\Omega): \mathcal I_\mu (u) <\rho\}$$ and $$\mathcal{B}^2_\rho= \big\{u \in W^{1,p}(\Omega): \Big(1 - \mu\frac{M_2}{pS_{p, \partial\Omega}}\Big)\|u\|^p_{1,p} <\rho\}.$$ Then $\mathcal{B}^1_\rho \subset \mathcal{B}^2_\rho$. From (\ref{e2.4}) we get $$\label{e2.5} |\mathcal J(u)| \leq \epsilon \Big(1 - \mu\frac{M_2}{pS_{p, \partial\Omega}}\Big)\|u\|^p_{1,p}+ \frac{C_{\delta}}{S^\frac{q}{p}_{q,\Omega}} \|u\|_{1,p}^q.$$ It is clear that $0 \in \mathcal{B}^1_\rho$ and $\mathcal J (0) = 0$. Hence, $0 \leq \sup_{u \in \mathcal{B}^1_\rho}\mathcal J (u)$, using (\ref{e2.5}) we get \begin{align}\label{e2.6} 0 \leq \frac{\sup_{u \in \mathcal{B}^1_\rho}\mathcal J(u)}{\rho} & \leq \frac{\sup_{u \in \mathcal{B}^2_\rho}\mathcal J(u)}{\rho} \leq \epsilon + \frac{C_{\delta}}{S^\frac{q}{p}_{q,\Omega}} \Big(1 - \mu\frac{M_2}{pS_{p, \partial\Omega}}\Big)^{-\frac{q}{p}} \rho^{\frac{q}{p}-1}. \end{align} We complete the proof of the lemma by letting $\rho \to 0^+$, since $\epsilon > 0$ is arbitrary. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.2} completed] Let $s_0$ be as in (H3). We choose a constant $r_0 >0$ such that $r_0 <$ dist$(0, \partial\Omega)$. For each $\delta \in (0,1)$ we define the function \begin{equation*} u_\delta(x) = \begin{cases} 0, & \text{if } x \in \mathbb{R}^N \backslash B_{r_0}(0) \\ s_0, & \text{if } x \in B_{\delta r_0}(0) \\ \frac{s_0}{r_0(1-\delta)} (r_0 - |x|), & \text{if } x \in B_{r_0}(0) \backslash B_{\delta r_0}(0), \end{cases} \end{equation*} where $B_{r_0}(0)$ denotes the open ball with center $0$ and radius $r_0 > 0$. Then, it is clear that $u_\delta \in W^{1,p}_0(\Omega)$. Moreover, we have \begin{gather}\label{e2.7} \|u_\delta\|^p_{1,p} \geq \frac{|s_0|^p(1-\delta^N)}{(1-\delta)^p}r_0^{N-p} \omega_N > 0,\\ \label{e2.8} \mathcal J (u_\delta) \geq [F(s_0)\delta^{N} - \max_{|t| \leq |s_0|}|F(t)|(1-\delta^{N})]\omega_{N}r_0^{N}, \end{gather} where $\omega_{N}$ is the volume of the unit ball in $\mathbb{R}^{N}$. From (\ref{e2.8}), there is $\delta_0 > 0$ such that $\|u_{\delta_0}\|_{1,p} > 0$ and $\mathcal J (u_{\delta_0}) > 0$. Now, by Lemma \ref{lem2.3}, we can choose $\rho_0 \in (0, 1)$ such that $$\rho_0 < \Big(1 - \mu\frac{M_2}{pS_{p, \partial\Omega}}\Big) \|u_{\delta_0}\|^p_{1,p} \leq \mathcal I_\mu (u_{\delta_0})$$ and satisfies $$\frac{\sup\{\mathcal J(u) : \mathcal I_\mu (u) < \rho_0\}}{\rho_0} < \frac{\mathcal J (u_{\delta_0})}{2 \mathcal I_\mu (u_{\delta_0})}.$$ To apply Lemma \ref{lem1.3}, we choose $x_1 = u_{\delta_0}$ and $x_0 = 0$. Then, the assumptions (i) and (ii) of Lemma \ref{lem1.3} are satisfied. Next, we define $$a_\mu = \frac{1+\rho_0}{\frac{\mathcal J (u_{\delta_0})}{ \mathcal I_\mu (u_{\delta_0})} - \frac{\sup\{\mathcal J (u) : \mathcal I_\mu (u) < \rho_0\}}{\rho_0}} > 0 \quad\text{and}\quad \overline{a}_\mu = \min\{a_\mu, \overline{\lambda}\}.$$ A simple computation implies that $(\bf {iii})$ are verified. Hence, there exist an open interval $\Lambda_\mu \subset [0, \overline{a}_\mu]$ and a real positive number $\delta_\mu$ such that for each $\lambda \in \Lambda_\mu$, the equation $D\Phi_{\mu,\lambda}(u) = D\mathcal I_{\mu}(u) - \lambda D \mathcal J(u) = 0$ has at least three solutions in $W^{1,p}(\Omega)$ whose $\|\cdot\|_{1,p}$-norms are less than $\delta_\mu$. By (H1) and (H2), one of them may be the trivial one. Thus, (\ref{e1.1}) has at least two weak solutions in $W^{1,p}(\Omega)$. The proof is complete. \end{proof} \subsection*{Acknowledgments} The author would like to thank the referees for their suggesions and helpful comments on this work. \begin{thebibliography}{00} \bibitem{1} E. A. M. Abreu, J. M. do \'{O} and E.S. Medeiros; Multiplicity of solutions for a class of quasilinear nonhomogeneuous Neumann problems, \emph{Nonlinear Analysis}, \textbf{60} (2005), 1443 - 1471. \bibitem{2} G. Bonanno; Some remarks on a three critical points theorem \emph{Nonlinear Analysis,} \textbf{54} (2003), 651 - 665. \bibitem{3} J. F. Bonder; Multiple positive solutions for quasilinear elliptic problems with sign-changing nonlinearities, \emph{Abstr. Appl. Anal.,} \textbf{2004}(12) (2004), 1047 - 1056. \bibitem{4} J. F. Bonder; Multiple solutions for the $p$-Laplacian equation with nonlinear boundary conditions, \emph{Electron. J. Diff. Eqns.,} \textbf{2006} no. 37, (2006), 1-7. \bibitem{5} J. F. Bonder, S. Mart\'{i}nez and J. D. Rossi; Existence results for gradient elliptic systems with nonlinear boundary conditions, \emph{NoDEA}, \textbf{14}(1-2) (2007), 153 - 179. \bibitem{6} J. F. Bonder and J. D. Rossi; Existence Results for the p-Laplacian with Nonlinear Boundary Conditions, \emph{J. Math. Anal. Appl.,} \textbf{263} (2001), 195 - 223. \bibitem{7} X. Fan; Boundary trace embedding theorems for variable exponent Sobolev spaces, \emph{J. Math. Anal. Appl.,} \textbf{339} (2008), 1395 - 1412. \bibitem{8} K. Perera; Multiple positive solutions for a class of quasilinear elliptic boundary-value problems, \emph{Electron. J. Diff. Eqns.,} \textbf{07} (2003), 1 - 5. \bibitem{9} B. Ricceri; On a three critical points, \emph{Arch. Math.,} (Basel) \textbf{75} (2000), 220 - 226. \bibitem{10} B. Ricceri; Existence of three solutions for a class of elliptic eigenvalue problems, \emph{Math. Comput. Modelling,} \textbf{32} (2000), 1485 - 1494. \bibitem{11} J. Yao; Solutions for Neumann boundary value problems involving $p(x)$-Laplace operators, \emph{Nonlinear Analysis,} \textbf{68} (2006), 1271 - 1283. \bibitem{12} Ji-Hong Zhao, Pei-Hao Zhao; Infinitely many weak solutions for a $p$-Laplacian equations with nonlinear boundary conditions, \emph{Electron. J. Diff. Eqns.,} \textbf{2006} no. 90 (2006), 1-14. \end{thebibliography} \end{document}